Manifold Diffusion Fields

We present Manifold Diffusion Fields (MDF), an approach that unlocks learning of diffusion models of data in general non-Euclidean geometries. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator. MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. In addition, we show that MDF generalizes to the case where the training set contains functions on different manifolds. Empirical results on multiple datasets and manifolds including challenging scientific problems like weather prediction or molecular conformation show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.